Aharoni-Pouzet基交换定理的推广
On generalisations of the Aharoni-Pouzet base exchange theorem
摘要：
Greene-Magnanti定理指出，如果$ M $是有限拟阵，$ B_0 $和$ B_1 $是基，$ B_0=\bigcup_{i=1}^{n} X_i $是一个划分，则存在一个划分$ B_1=\bigcup_{i=1}^{n}Y_i $使得$ (B_0 \setminus X_i) \cup Y_i $是每个$ i $的基。每个$ X_i $都是单例的特殊情况可以被重新表述为在基转换图中存在完美匹配。Pouzet强调，这仍然是真实的无限维向量空间。后来，他和Aharoni回答这个猜想肯定不只是向量空间，但无限拟阵。我们证明了他们的结果的两个推广。一方面，我们表明，“是一个单元素”可以放宽到“有限”，这是尖锐的意义上的无限集的排除是真正必要的。另一方面，我们证明了，如果$ B_0$和$ B_1 $是基地，那么有一个双射$ F $之间的有限子集，使$ (B_0\setminus I) \cup F(I) $是一个基地，每个$ I$。与Aharoni和Pouzet的方法相反，我们的证明是完全初级的，它们不依赖于无限匹配理论。
英文摘要：
The Greene-Magnanti theorem states that if $ M $ is a finite matroid, $ B_0 $ and $ B_1 $ are bases and $ B_0=\bigcup_{i=1}^{n} X_i $ is a partition, then there is a partition $ B_1=\bigcup_{i=1}^{n}Y_i $ such that $ (B_0 \setminus X_i) \cup Y_i $ is a base for every $ i $. The special case where each $ X_i $ is a singleton can be rephrased as the existence of a perfect matching in the base transition graph. Pouzet conjectured that this remains true in infinite dimensional vector spaces. Later he and Aharoni answered this conjecture affirmatively not just for vector spaces but for infinite matroids. We prove two generalisations of their result. On the one hand, we show that `being a singleton' can be relaxed to `being finite' and this is sharp in the sense the exclusion of infinite sets is really necessary. On the other hand, we prove that if $ B_0$ and $ B_1 $ are bases, then there is a bijection $ F $ between their finite subsets such that $ (B_0\setminus I) \cup F(I) $ is a base for every $ I$. In contrast to the approach of Aharoni and Pouzet, our proofs are completely elementary, they do not rely on infinite matching theory.
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